Pochhammer

The Pochhammer symbol $(a)_n$ is a notation that denotes the "rising factorial" function. It is defined by $$(a)_n = \left\{ \begin{array}{ll} 1, & \quad n=0 \\ \displaystyle\prod_{k=0}^{n-1} a+k=a(a+1)(a+2)\ldots(a+n-1), & \quad n=1,2,3,\ldots. \end{array} \right.$$

Properties

Sum of reciprocal Pochhammer symbols of a fixed exponent
Relationship between Pochhammer and gamma

Notes

We are using this symbol to denote the rising factorial (following the notation used by Abramowitz&Stegun and Mathematica) as opposed to denoting the falling factorial (as Wikipedia does).

References

• 1960: Earl David Rainville: Special Functions ... (previous) ... (next): $18. (1)$ (note: Rainville calls this the "factorial function" and expresses it slightly differently by defining it by the equivalent formula $(\alpha)_n=\displaystyle\prod_{k=1}^n (\alpha+k-1)$)
• 1964: W.N. Bailey: Generalized Hypergeometric Series ... (next): Section $1.1$